172 research outputs found
Euclidean Greedy Drawings of Trees
Greedy embedding (or drawing) is a simple and efficient strategy to route
messages in wireless sensor networks. For each source-destination pair of nodes
s, t in a greedy embedding there is always a neighbor u of s that is closer to
t according to some distance metric. The existence of greedy embeddings in the
Euclidean plane R^2 is known for certain graph classes such as 3-connected
planar graphs. We completely characterize the trees that admit a greedy
embedding in R^2. This answers a question by Angelini et al. (Graph Drawing
2009) and is a further step in characterizing the graphs that admit Euclidean
greedy embeddings.Comment: Expanded version of a paper to appear in the 21st European Symposium
on Algorithms (ESA 2013). 24 pages, 20 figure
On the Area Requirements of Planar Greedy Drawings of Triconnected Planar Graphs
In this paper we study the area requirements of planar greedy drawings of
triconnected planar graphs. Cao, Strelzoff, and Sun exhibited a family
of subdivisions of triconnected plane graphs and claimed that every planar
greedy drawing of the graphs in respecting the prescribed plane
embedding requires exponential area. However, we show that every -vertex
graph in actually has a planar greedy drawing respecting the
prescribed plane embedding on an grid. This reopens the
question whether triconnected planar graphs admit planar greedy drawings on a
polynomial-size grid. Further, we provide evidence for a positive answer to the
above question by proving that every -vertex Halin graph admits a planar
greedy drawing on an grid. Both such results are obtained by
actually constructing drawings that are convex and angle-monotone. Finally, we
consider -Schnyder drawings, which are angle-monotone and hence greedy
if , and show that there exist planar triangulations for
which every -Schnyder drawing with a fixed requires
exponential area for any resolution rule
Manhattan orbifolds
We investigate a class of metrics for 2-manifolds in which, except for a
discrete set of singular points, the metric is locally isometric to an L_1 (or
equivalently L_infinity) metric, and show that with certain additional
conditions such metrics are injective. We use this construction to find the
tight span of squaregraphs and related graphs, and we find an injective metric
that approximates the distances in the hyperbolic plane analogously to the way
the rectilinear metrics approximate the Euclidean distance.Comment: 17 pages, 15 figures. Some definitions and proofs have been revised
since the previous version, and a new example has been adde
Remarks on Category-Based Routing in Social Networks
It is well known that individuals can route messages on short paths through
social networks, given only simple information about the target and using only
local knowledge about the topology. Sociologists conjecture that people find
routes greedily by passing the message to an acquaintance that has more in
common with the target than themselves, e.g. if a dentist in Saarbr\"ucken
wants to send a message to a specific lawyer in Munich, he may forward it to
someone who is a lawyer and/or lives in Munich. Modelling this setting,
Eppstein et al. introduced the notion of category-based routing. The goal is to
assign a set of categories to each node of a graph such that greedy routing is
possible. By proving bounds on the number of categories a node has to be in we
can argue about the plausibility of the underlying sociological model. In this
paper we substantially improve the upper bounds introduced by Eppstein et al.
and prove new lower bounds.Comment: 21 page
Some Results on Greedy Embeddings in Metric Spaces
Geographic Routing is a family of routing algorithms that uses geographic point locations as addresses for the purposes of routing. Such routing algorithms have proven to be both simple to implement and heuristically effective when applied to wireless sensor networks. Greedy Routing is a natural abstraction of this model in which nodes are assigned virtual coordinates in a metric space, and these coordinates are used to perform point-to-point routing.
Here we resolve a conjecture of Papadimitriou and Ratajczak that every 3-connected planar graph admits a greedy embedding into the Euclidean plane. This immediately implies that all 3-connected graphs that exclude K 3,3 as a minor admit a greedy embedding into the Euclidean plane. We also prove a combinatorial condition that guarantees nonembeddability. We use this result to construct graphs that can be greedily embedded into the Euclidean plane, but for which no spanning tree admits such an embedding.Massachusetts Institute of Technology ((Akamai) Presidential Fellowship
Overlay Addressing and Routing System Based on Hyperbolic Geometry
International audienceLocal knowledge routing schemes based on virtual coordinates taken from the hyperbolic plane have attracted considerable interest in recent years. In this paper, we propose a new approach for seizing the power of the hyperbolic geometry. We aim at building a scalable and reliable system for creating and managing overlay networks over the Internet. The system is implemented as a peer-to-peer infrastructure based on the transport layer connections between the peers. Through analysis, we show the limitations of the Poincaré disk model for providing virtual coordinates. Through simulations, we assess the practicability of our proposal. Results show that peer-to-peer overlays based on hyperbolic geometry have acceptable performances while introducing scalability and flexibility in dynamic peer-to-peer overlay networks
Force-directed embedding of scale-free networks in the hyperbolic plane
Force-directed drawing algorithms are the most commonly used approach to visualize networks. While they are usually very robust, the performance of Euclidean spring embedders decreases if the graph exhibits the high level of heterogeneity that typically occurs in scale-free real-world networks. As heterogeneity naturally emerges from hyperbolic geometry (in fact, scale-free networks are often perceived to have an underlying hyperbolic geometry), it is natural to embed them into the hyperbolic plane instead. Previous techniques that produce hyperbolic embeddings usually make assumptions about the given network, which (if not met) impairs the quality of the embedding. It is still an open problem to adapt force-directed embedding algorithms to make use of the heterogeneity of the hyperbolic plane, while also preserving their robustness.
We identify fundamental differences between the behavior of spring embedders in Euclidean and hyperbolic space, and adapt the technique to take advantage of the heterogeneity of the hyperbolic plane
Drawing Graphs as Spanners
We study the problem of embedding graphs in the plane as good geometric
spanners. That is, for a graph , the goal is to construct a straight-line
drawing of in the plane such that, for any two vertices and
of , the ratio between the minimum length of any path from to
and the Euclidean distance between and is small. The maximum such
ratio, over all pairs of vertices of , is the spanning ratio of .
First, we show that deciding whether a graph admits a straight-line drawing
with spanning ratio , a proper straight-line drawing with spanning ratio
, and a planar straight-line drawing with spanning ratio are
NP-complete, -complete, and linear-time solvable problems,
respectively, where a drawing is proper if no two vertices overlap and no edge
overlaps a vertex.
Second, we show that moving from spanning ratio to spanning ratio
allows us to draw every graph. Namely, we prove that, for every
, every (planar) graph admits a proper (resp. planar) straight-line
drawing with spanning ratio smaller than .
Third, our drawings with spanning ratio smaller than have large
edge-length ratio, that is, the ratio between the length of the longest edge
and the length of the shortest edge is exponential. We show that this is
sometimes unavoidable. More generally, we identify having bounded toughness as
the criterion that distinguishes graphs that admit straight-line drawings with
constant spanning ratio and polynomial edge-length ratio from graphs that
require exponential edge-length ratio in any straight-line drawing with
constant spanning ratio
- âŠ